Really Struggling :(

Hi,
I am really struggling with a subject at my university which is all about partial differential equations. It takes me a long time to understand even the most basic concepts, and I really need some help understanding and completing the following questions.

Hi,
I am really struggling with a subject at my university which is all about partial differential equations. It takes me a long time to understand even the most basic concepts, and I really need some help understanding and completing the following questions.

Let be a bounded region in , and suppose on .

(i) If is a solution of

show that on

(ii) If is a solution of

show that is unique (It can be assumed that part (i) is true).

Well, it looks like a typical application of the Lax-Milgram theorem to me.

The weak formulation of this problem on reads

or, by integrating the left hand side by parts and using the boundary conditions,

.

Define the bilinear form on ,

.

To show that is solvable for some , we can invoke the Lax-Milgram theorem: We need just show that is bounded and coercive.

You can show it is bounded, ie there exists a such that

, where

As for being coercive, we must show there is with . We compute

.

Now the Lax-Milgram theorem applies, and there is a such that .
And reflexivity implies .

For part ii), consider two solutions of the problem at hand.
Then their difference is a solution for the problem given in part i), and therefore must be zero.

Ps. You cannot construct a Green's function here, as zero is an eigenvalue for on a region.